Fully-characteristic congruence

A congruence $ \theta $ of an algebraic system $ A = \langle A, \Omega \rangle $ which is invariant under any endomorphism $ \sigma $ of this system, i.e. it follows from $ x \theta y $ that $ \sigma ( x) \theta \sigma ( y) $( $ x, y \in A $). The fully-characteristic congruences of an algebraic system $ A $ form under inclusion a complete sublattice $ C _ {v} ( A) $ of the lattice $ C( A) $ of all congruences of $ A $. If $ \mathfrak M $ is a variety of $ \Omega $- systems and if $ F $ is a free algebra in $ \mathfrak M $ on a countably infinite set of generators, the lattice $ C _ {v} ( F) $ of fully-characteristic congruences of the system $ F $ is dually isomorphic to the lattice $ L _ {v} ( \mathfrak M ) $ of all subvarieties of $ \mathfrak M $. Any congruence $ \kappa $ of an $ \Omega $- algebra $ A $ with a finite number of generators, of finite index in $ A $( i.e. with a finite number of congruence classes $ a/ \kappa $, $ a \in A $), contains a fully-characteristic congruence $ \theta $ of $ A $ which also has finite index in $ A $.

References

[1]	A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)

Comments

Fully-characteristic congruences are also called fully-invariant congruences.

References

[a1]	P.M. Cohn, "Universal algebra" , Reidel (1981)